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Fast Computation of Bernoulli, Tangent and Secant Numbers

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Computational and Analytical Mathematics

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 50))

Abstract

We consider the computation of Bernoulli, Tangent (zag), and Secant (zig or Euler) numbers. In particular, we give asymptotically fast algorithms for computing the first n such numbers O(n 2(logn)2 + o(1)). We also give very short in-place algorithms for computing the first n Tangent or Secant numbers in O(n 2) integer operations. These algorithms are extremely simple and fast for moderate values of n. They are faster and use less space than the algorithms of Atkinson (for Tangent and Secant numbers) and Akiyama and Tanigawa (for Bernoulli numbers).

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Acknowledgements

We thank Jon Borwein for encouraging the belief that high-precision computations are useful in “experimental” mathematics [4], e.g. in the PSLQ algorithm [16]. Ben F. “Tex” Logan, Jr. (unpublished observation, mentioned in [18, Sect. 6.5]) suggested the use of Tangent numbers to compute Bernoulli numbers. Christian Reinsch (about 1979, acknowledged in [6], Personal communication to R.P. Brent) pointed out the numerical instability of the recurrence (8.7) and suggested the use of the numerically stable recurrence (8.8). Christopher Heckman kindly drew our attention to Atkinson’s algorithm [2]. We thank Paul Zimmermann for his comments. Some of the material presented here is drawn from the recent book Modern Computer Arithmetic [7] (and as-yet-unpublished solutions to exercises in the book). In particular, see [7, Sect. 4.7.2 and Exercises 4.35–4.41]. Finally, we thank David Bailey, Richard Crandall, and two anonymous referees for suggestions and pointers to additional references.

Author information

Authors and Affiliations

  1. Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia

    Richard P. Brent

  2. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    David Harvey

Authors
  1. Richard P. Brent
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  2. David Harvey
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Corresponding author

Correspondence to Richard P. Brent .

Editor information

Editors and Affiliations

  1. Lawrence Berkeley National Laboratory, Berkeley, USA

    David H. Bailey

  2. Mathematics, University of British Columbia, Kelowna, British Columbia, Canada

    Heinz H. Bauschke

  3. Dept. Mathematics & Statistics, Simon Fraser University, Burnaby, British Columbia, Canada

    Peter Borwein

  4. Mathematics, University of Florida, Gainesville, Florida, USA

    Frank Garvan

  5. Mathematics and Computer Science, University of Limoges, Limoges, France

    Michel Théra

  6. Mathematics and Computer Science Department, La Sierra University, Riverside, California, USA

    Jon D. Vanderwerff

  7. Dept. Combinatorics & Optimization, University of Waterloo Faculty of Mathematics, Waterloo, Ontario, Canada

    Henry Wolkowicz

Additional information

Note added in proof

Recently the second author [A subquadratic algorithm for computing the n-th Bernoulli number, arXiv:1209.0533, to appear in Mathematics of Computation] has given an improved algorithm for the computation of a single Bernoulli number. The new algorithm reduces the exponent from 2 + o(1) to \(4/3 + o(1)\).

Communicated by David H. Bailey.

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Brent, R.P., Harvey, D. (2013). Fast Computation of Bernoulli, Tangent and Secant Numbers. In: Bailey, D., et al. Computational and Analytical Mathematics. Springer Proceedings in Mathematics & Statistics, vol 50. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7621-4_8

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